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CONSTRUCTION OF COMPACTLY SUPPORTED WAVELETS FROM TRIGONOMETRIC B-SPLINES

INTERNATIONAL JOURNAL OF WAVELETS MULTIRESOLUTION AND INFORMATION PROCESSING 2011年第5期9卷 843-865页

作者： Jena, M. K. VSS Univ Technol Dept Math Burla 768018 Orissa India

We construct a class of semiorthogonal wavelets by taking a normalized trigonometric B-spline of any order as the scaling function. The construction is based on generalized Euler-Frobenius polynomial and generalized autocorrelation function. We also show that the odd order normalized trigonometric B-spline satisfies convex hull property as well as partition of unity property. Moreover, we also present a subdivision algorithm for the convolution of normalized trigonometric B-splines. Several examples of wavelet are also provided.

关键词： Trigonometric B-splines multiresolution analysis wavelet Euler-Frobenius polynomial autocorrelation function subdivision algorithm Poisson's sum

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Real-time smooth trajectory generation for nonholonomic mobile robots using Bezier curves

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ROBOTICS AND COMPUTER-INTEGRATED MANUFACTURING 2016年 41卷 31-42页

作者： Simba, Kenneth Renny Uchiyama, Naoki Sano, Shigenori Toyohashi Univ Technol Dept Mech Engn Toyohashi Aichi 4418580 Japan Tanzania Atom Energy Commiss Directorate Nucl Technol Arusha Tanzania

This paper focuses on generating smooth trajectories for a wheeled nonholonomic mobile robot using piecewise Bezier curves with properties ideally suited for this purpose. The developed algorithm generates smooth motion trajectories with C-2 continuous curvature. We consider a teleoperated wheeled mobile robot in an indoor environment with ceiling cameras for operator visibility. The motion trajectory is constrained by the operator-specified via points and path width. A method to automatically generate a trajectory based on only these two inputs is proposed and demonstrated. To improve the trackability of the mobile robot, we adopted a Bezier subdivision method and inserted a quintic Bezier segment into high-curvature areas. The proposed algorithm can be used for real-time obstacle-avoidance trajectory generation because it allows trajectory subdivision and arbitrarily setting of the second derivative at the start point. Simulation and experimental results demonstrate the effectiveness of the proposed method. (C) 2016 Elsevier Ltd. All rights reserved.

关键词： Bezier curves Continuous curvature Mobile robot Motion planning Motion trajectory subdivision algorithm

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Efficient approximation of minimum energy curves with interpolatory constraints

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APPLIED MATHEMATICS AND COMPUTATION 2000年第2-3期109卷 151-166页

作者： Qu, RB Ye, JP Natl Univ Singapore Dept Math Singapore 119260 Singapore

Different methods for the approximation of a set of data points with interpolatory property and appropriate boundary conditions are investigated with respect to the exact energy value. It is found that for a given set of data points on a plane, the 6-point interpolatory subdivision method could be the best choice among the current widely used methods such as cubic splines and exponential splines due to its simplicity, locality, efficiency and most of all, its near-minimum energy property. Examples and graphics are provided to show these properties of the curves produced by the subdivision algorithm. (C) 2000 Elsevier Science Inc. All rights reserved.

关键词： minimal energy curve spline subdivision algorithm interpolation approximation

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Spline interpolation and wavelet construction

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APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 1998年第3期5卷 249-276页

作者： Lee, SL Sharma, A Tan, HH Natl Univ Singapore Dept Math Singapore 119260 Singapore Univ Alberta Dept Math Edmonton AB T6G 2G1 Canada

The method of Dubuc and Deslauriers on symmetric interpolatory subdivision is extended to study the relationship between interpolation processes and wavelet construction. Refinable and interpolatory functions are constructed in stages from B-splines. Their method constructs the filter sequence (its Laurent polynomial) of the interpolatory function as a product of Laurent polynomials. This provides a natural way of splitting the filter for the construction of orthonormal and biorthogonal scaling functions leading to orthonormal and biorthogonal wavelets. Their method also leads to a class of filters which includes the minimal length Daubechies compactly supported orthonormal wavelet coefficients. Examples of ''good'' filters are given together with results of numerical experiments conducted to test the performance of these filters in data compression. (C) 1998 Academic Press.

关键词： refinable function wavelets interpolatory function uniform B-spline Euler-Frobenius polynomial Riesz basis biorthogonal basis orthonormal basis transition operator subdivision algorithm cascade algorithm condition E

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An algorithm for finding all solutions of a nonlinear system

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2001年第2期137卷 293-315页

作者： Smiley, MW Chun, CB Iowa State Univ Dept Math Ames IA 50011 USA Seoul Natl Univ Dept Math Seoul South Korea

Let f : X --> R-k be a Lipschitz continuous function on a compact subset X subset of R-d. subdivision algorithms are described that can be used to find all solutions of the equation f(x) = 0 that lie in X, Convergence is shown and numerical examples are presented. Modifications of the basic algorithm which speed convergence are given for the case of nondegenerate zeros of a vector field. (C) 2001 Elsevier Science B.V. All rights reserved.

关键词： subdivision algorithm Newton's method

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From curve design algorithms to the design of rigid body motions

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VISUAL COMPUTER 2004年第5期20卷 279-297页

作者： Hofer, M Pottmann, H Ravani, B Vienna Univ Technol Geometr Modeling & Ind Geometry Grp A-1040 Vienna Austria Univ Calif Davis Dept Mech & Aeronaut Engn Davis CA 95616 USA

We discuss the following problem, which arises in computer animation and robot motion planning: given are N positions or keyframes Sigma(t(i)) of a moving body Sigma subset of R-3 at time instances ti. Compute a smooth rigid body motion Sigma(t) that interpolates or approximates the given positions Sigma(t(i)) such that chosen feature points of the moving system run on smooth paths. We present an algorithm that can be considered as a transfer principle from curve design algorithms to motion design. The algorithm relies on known curve design algorithms and on registration techniques from computer vision. We prove that the motion generated in this way is of the same smoothness as the curve design algorithm employed.

关键词： motion design motion planning registration subdivision algorithm variational design

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SqFreeEVAL: An (almost) optimal real-root isolation algorithm

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JOURNAL OF SYMBOLIC COMPUTATION 2012年第2期47卷 153-166页

作者： Burr, Michael A. Krahmer, Felix Fordham Univ Bronx NY 10458 USA Univ Bonn Hausdorff Ctr Math D-53115 Bonn Germany NYU Courant Inst Math Sci New York NY 10012 USA

Let f be a univariate polynomial with real coefficients, f is an element of R[X]. subdivision algorithms based on algebraic techniques (e.g., Sturm or Descartes methods) are widely used for isolating the real roots of f in a given interval. In this paper, we consider a simple subdivision algorithm whose primitives are purely numerical (e.g., function evaluation). The complexity of this algorithm is adaptive because the algorithm makes decisions based on local data. The complexity analysis of adaptive algorithms (and this algorithm in particular) is a new challenge for computer science. In this paper, we compute the size of the subdivision tree for the SqFreeEVAL algorithm. The SqFreeEVAL algorithm is an evaluation-based numerical algorithm which is well-known in several communities. The algorithm itself is simple, but prior attempts to compute its complexity have proven to be quite technical and have yielded sub-optimal results. Our main result is a simple O(d(L+ In d)) bound on the size of the subdivision tree for the SqFreeEVAL algorithm on the benchmark problem of isolating all real roots of an integer polynomial f of degree d and whose coefficients can be written with at most L bits. Our proof uses two amortization-based techniques: first, we use the algebraic amortization technique of the standard Mahler-Davenport root bounds to interpret the integral in terms of d and L. Second, we use a continuous amortization technique based on an integral to bound the size of the subdivision tree. This paper is the first to use the novel analysis technique of continuous amortization to derive state of the art complexity bounds. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： Continuous amortization Adaptive analysis subdivision algorithm Integral analysis Amortization Root isolation

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Condition numbers for the cube. I: Univariate polynomials and hypersurfaces

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JOURNAL OF SYMBOLIC COMPUTATION 2023年 115卷 142-173页

作者： Tonelli-Cueto, Josue Tsigaridas, Elias Sorbonne Univ Inria Paris & IMJ PRG 4 Pl Jussieu F-75005 Paris France

The condition-based complexity analysis framework is one of the gems of modern numerical algebraic geometry and theoretical computer science. Among the challenges that it poses is to expand the currently limited range of random polynomials that we can handle. Despite important recent progress, the available tools cannot handle random sparse polynomials and Gaussian polynomials, that is polynomials whose coefficients are i.i.d. Gaussian random variables. We initiate a condition-based complexity framework based on the norm of the cube that is a step in this direction. We present this framework for real hypersurfaces and univariate polynomials. We demonstrate its capabilities in two problems, under very mild probabilistic assumptions. On the one hand, we show that the average run-time of the Plantinga-Vegter algorithm is polynomial in the degree for random sparse (alas a restricted sparseness structure) polynomials and random Gaussian polynomials. On the other hand, we study the size of the subdivision tree for Descartes' solver and run-time of the solver by Jindal and Sagraloff (2017). In both cases, we provide a bound that is polynomial in the size of the input (size of the support plus the logarithm of the degree) not only for the average but also for all higher moments. (c) 2022 Elsevier Ltd. All rights reserved.

关键词： Condition number Random polynomial subdivision algorithm Univariate solver

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Quadratic-attraction subdivision with contraction-ratio λ=1/2

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COMPUTERS & GRAPHICS-UK 2024年 123卷

作者： Karciauskas, Kestutis Peters, Jorg Vilnius Univ Inst Math Vilnius Lithuania Univ Florida Dept CISE Gainesville FL 32611 USA

Classic generalized subdivision, such as Catmull-Clark subdivision, as well as recent subdivision algorithms for high-quality surfaces, rely on slower convergence towards extraordinary points for mesh nodes surrounded by n > 4 quadrilaterals. Slow convergence corresponds to a contraction-ratio of ) > 0 . 5 . To improve shape, prevent parameterization discordant with surface growth, or to improve convergence in isogeometric analysis near extraordinary points, a number of algorithms explicitly adjust ) by altering refinement rules. However, such tuning of ) has so far led to poorer surface quality, visible as uneven distribution or oscillation of highlight lines. The recent Quadratic-Attraction subdivision (QAS) generates high-quality, bounded curvature surfaces based on a careful choice of quadratic expansion at the central point and, just like Catmull-Clark subdivision, creates the control points of the next subdivision ring by matrix multiplication. But QAS shares the contraction- ratio ) lambda(CC) > 1/2 of Catmull-Clark subdivision when n > 4. For n = 5 , ... , 10, , QAS, improves the convergence to the uniform ) lambda = 1/2 of binary domain refinement and without sacrificing surface quality compared to QAS.

关键词： subdivision algorithm Surface quality Quadratic expansion Uniform contraction Curvature

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Voronoi diagrams of algebraic distance fields

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COMPUTER-AIDED DESIGN 2013年第2期45卷 511-516页

作者： Emiris, Ioannis Z. Mantzaflaris, Angelos Mourrain, Bernard Austrian Acad Sci RICAM Linz Linz Austria Univ Athens Dept Informat & Telecommun Athens Greece INRIA Mediterranee GALAAD Sophia Antipolis France

We design and implement an efficient and certified algorithm for the computation of Voronoi diagrams (VDs) constrained to a given domain. Our framework is general and applicable to any VD-type where the distance field is given explicitly or implicitly by a polynomial, notably the anisotropic VD or VDs of non-punctual sites. We use the Bernstein form of polynomials and DeCasteljau's algorithm to subdivide the initial domain and isolate bisector, or domains that contain a Voronoi vertex. The efficiency of our algorithm is due to a filtering process, based on bounding the field over the subdivided domains. This allows to exclude functions (thus sites) that do not contribute locally to the lower envelope of the lifted diagram. The output is a polygonal description of each Voronoi cell, within any user-defined precision, isotopic to the exact VD. Correctness of the result is implied by the certified approximations of bisector branches, which are computed by existing methods for handling algebraic curves. First experiments with our C++ implementation, based on double precision arithmetic, demonstrate the adaptability of the algorithm. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Voronoi diagram subdivision algorithm Bisector curve Anisotropic diagram Lower envelope

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